One of the important goals of a sixth grade math curriculum is the beginning of the transition from arithmetic to algebra. To that end, we spend a lot of time talking about some different ways of thinking about problem solving.
Algebra gives us a distinctive way of solving problems, a deductive method with a twist. That twist is "backward thinking." Consider the problem of taking the number 25, adding 17 to it, and getting 42. This is "forward thinking." We're given the numbers and we just add them together. But, instead, suppose we were given the answer 42 and asked a different question. We now want the number which when added to 25 gives us 42. This is where backward thinking comes in. We want the value of x which solves the equation 25 + x = 42 and we subtract 25 from 42 to give it to us. (Working from back to front.)
Word problems which are meant to be solved by algebra have been given to students for centuries. I'm sure we all remember the two cars travelling toward each other at different speeds and having to determine when they would meet. Or the one that says:
My niece Michelle is 6 years old, and my age is 40 . When will I be three times as old as her?
This could be solved by trial and error, but algebra is more economical. In x years from now Michelle will be 6 + x years old and I will be 40 + x. I will be three times as old as her when 3 * (6 + x) = 40 + x.
Multiply out the left hand side of the equation and you'll get 18 + 3x = 40 + x, and by moving all the x's over to one side of the equation and the numbers to the other, we find that 2x = 22 which means that x = 11. When I'm 51 Michelle will be 17 years old... Magic! (These kinds of equations are called linear equations. They have no exponents and when plotted on a graph will create a straight line.)
Mathematics underwent a big change when it passed from the science of arithmetic to the science of symbols, or algebra. To progress from numbers to letters is a mental jump but the effort is certainly worthwhile.
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